We have $\sim$ relation of equivalence and $=$. We also know that there are infinite number of finite classes of equivalence. Does the theory with signature $(\sim, =)$ admits QE?
My answer is no, but I don't know how to prove it strong. We can make an infinite number of equivalence classes of size 1 and one class of size two. Then predicate $P(x) = \exists z (x\sim z \land x\ne z)$. It's true for class of size two and false for others, but I don't know how to prove it correctly.